Existence results for a class of nonlocal problems involving \(p(x)\)-Laplacian
type: Item
instance of: scholarly article
title: Existence results for a class of nonlocal problems involving \(p(x)\)-Laplacian
zbMATH Open document ID: 1317.35063
author: Mostafa Allaoui
published in: Arabian Journal of Mathematics
publication date: 2015-03-25T00:00:00Z
review text: The author finds sufficient conditions so that the boundary value problems below admit nontrivial weak solutions. The equations considered are \[ -M \Delta_{p(x)} u = \lambda \left( a(x)|u|^{q(x)-2}u+b(x) |u|^{r(x)-2}u\right) \;\text{ in } \Omega, \;\;\;u = 0 \text{ on } \partial \Omega. \] Here \(\Delta_{p(x)} u \) is the variable exponent \(p(x)\)-Laplacian, and \[ M = M\left( \int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)} \;dx\right) \] with \(M : \mathbb{R}^+\to \mathbb{R}\) a function such as for example \(M(t) = t\arctan(t)\). The existence of weak solutions is established via the Mountain Pass Theorem.
reviewed by: Florin Catrina
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J48
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 6419560
zbMATH Keywords: \(p(x)\)-Kirchhoff problem
zbMATH Keywords: weak solutions
zbMATH Keywords: Mountain Pass Theorem
MaRDI profile type: Publication
full work available at URL: s40065-014-0115-4
OpenAlex ID: W1823085969
Wikidata QID: Q59424739
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DOI: 10.1007/S40065-014-0115-4
description: scientific article; zbMATH DE number 6419560